Fidelity

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Quantum Fidelity

Fidelity is a popular measure of distance between density operators. It is not a metric, but has some useful properties and it can be used to defined a metric on this space of density matrices, known as Bures metric.

Fidelity as a distance measure between pure states used to be called "transition probability". For two states given by unit vectors φ,ψ it is .
For a pure state (vector ψ) and a mixed state (density matrix ρ) this generalizes to , and for two density matrices ρ,σ it is generalized as the largest fidelity between any two purifications of the given states. According to a theorem by Uhlmann, this leads to the expression

This is precisely the expression used by Richard Jozsa in<bibref>Jozsa94</bibref>, where the term fidelity appears to have been used first.

However, one can also start from , leading to the alternative

used in<bibref>NielsenChuang</bibref>. This second quantity is sometimes denoted as and called square root fidelity. It has no interpretation as a probability, but appears in some estimates in a simpler way.

Basic properties

If is pure, then and if both states are pure i.e. and , then .

Other properties:

Bures distance

Fidelity can be used to define metric on the set of quantum states, so called Bures distance<bibref>fuchs96phd</bibref> DB

and the angle<bibref>NielsenChuang</bibref>

The quantity DB(ρ,σ) is the minimal distance between purifications of ρ and σ using a common environment.

Classical fidelity

Fidelity is also defined for classical probability distributions. Let {pi} and {qi}
where i = 1,2,...,n be probability distributions. The fidelity between p and q
is defined as